A multidimensional generalization of Symanzik polynomials

TL;DR

This paper generalizes Symanzik polynomials from Feynman graphs to higher dimensional simplicial complexes, establishing dualities, geometric invariants, and stability results, with potential applications in quantum field theory and mathematics.

Contribution

It introduces a multidimensional generalization of Symanzik polynomials, explores their properties, dualities, and invariants, and extends the framework to matroids over hyperfields.

Findings

Established duality between generalized Symanzik and Kirchhoff polynomials.

Derived geometric invariants independent of triangulation.

Proved a stability theorem extending Amini's result to higher dimensions.

Abstract

Symanzik polynomials are defined on Feynman graphs and they are used in quantum field theory to compute Feynman amplitudes. They also appear in mathematics from different perspectives. For example, recent results show that they allow to describe asymptotic limits of geometric quantities associated to families of Riemann surfaces. In this paper, we propose a generalization of Symanzik polynomials to the setting of higher dimensional simplicial complexes and study their basic properties and applications. We state a duality relation between these generalized Symanzik polynomials and what we call Kirchhoff polynomials, which have been introduced in recent generalizations of Kirchhoff's matrix-tree theorem to simplicial complexes. Moreover, we obtain geometric invariants which compute interesting data on triangulable manifolds. As the name indicates, these invariants do not depend on the chosen triangulation. We furthermore prove a stability theorem concerning the ratio of Symanzik polynomials which extends a stability theorem of Amini to higher dimensional simplicial complexes. In order to show that theorem, we will make great use of matroids, and provide a complete classification of the connected components of the exchange graph of a matroid, a graph which encodes the exchange properties between independent sets of the matroid. We hope that this result could be of independent interest. Finally, we explain how to generalize Symanzik polynomials to the setting of matroids over hyperfields defined recently by Baker and Bowler.